Linking Cubes
Description
Linking Cubes are hands-on manipulatives that are used to assist with the understanding of mathematical concepts. Linking cubes are individual unit cubes that can interlock together to build various shapes and structures. They are user friendly for all ages and are a unique visualization tool. This manipulative is commercially sold in a variety of colours under brand names, such as Cube-A-Link.
Purpose
There are a number of reasons why I would use this manipulative. Firstly, it can help concretize learning. It does this by helping math become real and engaging, therefore making it more fun to learn for the students. Secondly, it causes abstract concepts to be more visual and physical. All of this aids in the development mathematical intuition (or math sense).
You can use linking cubes to meet expectations in strands such as:
Strand / ExamplePatterning and Algebra / growth patterns, linear models
Number Sense and Numeration / fractions, ratios, integers, factors
Geometry and Spatial Sense / isometric drawings, orthographic drawings, spatial sense
Measurement / Volume and surface area (3D), perimeter and area (2D)
Sample Activities
Activity 1: Linear Models - Growing L’s
This activity should be reserved for initiating the topic of linear models. First, provide a sequence on Isometric Dot Paper to your students and ask them to build a replicate of this sequence with a partner.
Once they have built the sequence they should be told to look for patterns both physical and numerical. You should provide the students with the chart below to help them work through the questions they are about to receive.
Model Number / 1 / 2 / 3 / 4Number of Cubes / 1 / 1 + 2
= 1 + 2
= 1 + (1) 2
= 3 / 3 + 2
= 1 + 2 + 2
= 1 + (2) 2
= 5 / 5 + 2
= 1 + 2 + 2 + 2
= 1 + (3) 2
= 7
The following questions should be used along with demonstrations to help students understand the numerical representations of the physical patterns they see.
Question 1: In the row “Number of Cubes”, you will see the sequence 1, 1+2, 3+2, and 5+2. Explain how these expressions are connected to the above sequence of models.
Answer: 2 cubes are added at each step
Question 2: How many Cubes will there be in the 5th, 6th, and 7th models?
Answer: Model 5 has 9 cubes, model 6 has 11 cubes and model 7 has 13 cubes
A cube is added to the top and front of each model
Question 3: In the row “Number of Cubes”, you will see the sequence 1, 1+2, 1+2+2, 1+2+2+2. Explain how these expressions are connected to the sequence of models.
Answer: The expressions illustrate how many times the 2 cubes has been added
Question 4: In the row “Number of Cubes”, how are the numbers in the brackets related to the model number?
Answer: The number in the brackets is one less than the model number
Questions 5: What number would be in the brackets for the 20th model? For the nth model?
Answer: The number in brackets for the 20th model would be 19. For the nth model would be n-1
Question 6: How many cubes would there be in the 45th model?
Answer: There would be 89 cubes in the 45th model.
Next, be sure to perform a follow-up analysis to summarize the finding from the above questions.
The number of cubes in the nth model is given by the formula:
N = 1 + (n-1) · 2
N = 2n - 1
Here, N represents the total number of cubes in the nth model. If you look carefully you will see that this formula reflects the way the models were constructed.
Question 7: In the formula, N = 1 + (n-1) · 2. We know that the expression (n-1) is connected to the model number but what about the “1” and the “2”?
Answer: The “1” in the formula was the number of cubes in the first model.
The “2” in the formula was the number of cubes added at each step
Activity 2: Linear Models – Sprouting Arms
The following sequence of models forms a definite pattern. Ask your students to build a replicate of this sequence with a partner. Have students describe the process that was used to form each successive model out of the previous one.
Once they have built the sequence they should be told to look for patterns both physical and numerical. The following questions should be used along with demonstrations to help students understand the numerical representations of the physical patterns they see.
Question 1: How many cubes would you add to the 4th model to create the 5th model? How many cubes would there be in the 5th model in the sequence?
Answer: 3 cubes are added to the 4th model to create the 5th model and there would be 13 cubes in the 5th model.
Question 2: Complete the following table with your prediction of the number of cubes in the 5th model and the formula that will give the number of cubes in the nth model.
Model Number / 1 / 2 / 3 / 4 / 5 / … / nNumber of Cubes
Answer:
Model Number / 1 / 2 / 3 / 4 / 5 / … / nNumber of Cubes / 1 / 4 / 7 / 10 / 13 / … / N=3n-2
For advanced learning:
Question 3: Will there be a model in this sequence that has exactly 71 cubes in it? Explain your thinking.
Answer: Since the equations 3n-2=71 has no solution in the natural numbers. There is no model with 71 cubes in it
Activity 3: Linear Models – Plus Pluses
The following sequence of models forms a definite pattern. Ask your students to build a replicate of this sequence with a partner. Have students describe the process that was used to form each successive model out of the previous one.
Once they have built the sequence they should be told to look for patterns both physical and numerical. The following questions should be used along with demonstrations to help students understand the numerical representations of the physical patterns they see.
Question 1: How many cubes would you add to the 4th model to create the 5th model? How many cubes would there be in the 5th model in the sequence?
Answer: 4 cubes are added to the 4th model to create the 5th model and there would be 17 cubes in the 5th model.
Question 2: Complete the following table with your prediction of the number of cubes in the 5th model and the formula that will give the number of cubes in the nth model.
Model Number / 1 / 2 / 3 / 4 / 5 / … / nNumber of Cubes
Answer:
Model Number / 1 / 2 / 3 / 4 / 5 / … / nNumber of Cubes / 1 / 5 / 9 / 13 / 17 / … / N=4n-3
Question 3: Consider the surface area of each model in the sequence of models. For each cube that is added at each step how many new “faces” are added to the total surface area of the model? Be careful with this question. Make sure that you are counting only additional faces.
Answer: 16 faces are added to the total surface area at each step. (4 new faces with each new cube)
Question 4: Complete the following table with your prediction of the number of faces on the surface of the 5th and 6th models. Can you find the formula that will give you the number of faces on the surface of the nth model?
Model Number / 1 / 2 / 3 / 4 / 5 / … / nNumber of Faces
Answer:
Model Number / 1 / 2 / 3 / 4 / 5 / … / nNumber of Faces / 6 / 22 / 38 / 54 / 70 / … / 16n-10
Black Line Masters
Activity 1: Linear Models - Growing L’s
Build this sequence with a partner using your Linking Cubes:
Look for patterns both physical and numerical.
Model Number / 1 / 2 / 3 / 4Number of Cubes / 1 / 1 + 2
= 1 + 2
= 1 + (1) 2
= 3 / 3 + 2
= 1 + 2 + 2
= 1 + (2) 2
= 5 / 5 + 2
= 1 + 2 + 2 + 2
= 1 + (3) 2
= 7
Using the chart above answer the following questions:
1. In the row “Number of Cubes”, you will see the sequence 1, 1+2, 3+2, and 5+2. Explain how these expressions are connected to the above sequence of models.
2. How many Cubes will there be in the 5th, 6th, and 7th models?
3. In the row “Number of Cubes”, you will see the sequence 1, 1+2, 1+2+2, 1+2+2+2. Explain how these expressions are connected to the sequence of models.
4. In the row “Number of Cubes”, how are the numbers in the brackets related to the model number?
5. What number would be in the brackets for the 20th model? For the nth model?
6. How many cubes would there be in the 45th model?
The number of cubes in the nth model is given by the formula:
N = 1 + (n-1) · 2
N = 2n - 1
Here, N represents the total number of cubes in the nth model.
7. In the formula, N = 1 + (n-1) · 2. We know that the expression (n-1) is connected to the model number but what about the “1” and the “2”?
Activity 2: Linear Models – Sprouting Arms
The following sequence of models forms a definite pattern. Build the following sequence. Describe the process that was used to form each successive model out of the previous one.
1. How many cubes would you add to the 4th model to create the 5th model? How many cubes would there be in the 5th model in the sequence?
2. Complete the following table with your prediction of the number of cubes in the 5th model and the formula that will give the number of cubes in the nth model.
Model Number / 1 / 2 / 3 / 4 / 5 / … / nNumber of Cubes
3. Will there be a model in this sequence that has exactly 71 cubes in it? Explain your thinking.
Activity 3: Linear Models – Plus Pluses
The following sequence of models forms a definite pattern. Build a replicate of this sequence with a partner. Describe the process that was used to form each successive model out of the previous one.
1. How many cubes would you add to the 4th model to create the 5th model? How many cubes would there be in the 5th model in the sequence?
2. Complete the following table with your prediction of the number of cubes in the 5th model and the formula that will give the number of cubes in the nth model.
Model Number / 1 / 2 / 3 / 4 / 5 / … / nNumber of Cubes
3. Consider the surface area of each model in the sequence of models. For each cube that is added at each step how many new “faces” are added to the total surface area of the model? Be careful with this question. Make sure that you are counting only additional faces.
4. Complete the following table with your prediction of the number of faces on the surface of the 5th and 6th models. Can you find the formula that will give you the number of faces on the surface of the nth model?
Model Number / 1 / 2 / 3 / 4 / 5 / … / nNumber of Faces